BLDC

Description

The BLDC block models a permanent magnet synchronous
machine with a three-phase wye-wound stator. The block has four options for defining the
permanent magnet flux distribution as a function of rotor angle. Two options allow for
simple parameterization by assuming a perfect trapezoid for the back emf. For simple
parameterization, you specify either the flux linkage or the rotor-induced back emf. The
other two options give more accurate results using tabulated data that you specify. For
more accurate results, you specify either the flux linkage partial derivative or the
measured back emf constant for a given rotor speed.

The figure shows the equivalent electrical circuit for the stator windings.

Motor Construction

This figure shows the motor construction with a single pole-pair on the rotor.

For the axes convention in the preceding figure, the
a-phase and permanent magnet fluxes are aligned when rotor
angle θr is zero. The block supports a
second rotor-axis definition. For the second definition, the rotor angle is the
angle between the a-phase magnetic axis and the rotor
q-axis.

Trapezoidal Rate of Change of Flux

The rotor magnetic field due to the permanent magnets create a trapezoidal rate of
change of flux with rotor angle. The figure shows this rate of change of
flux.

Back emf is the rate of change of flux, defined by

dΦdt=∂Φ∂θdθdt=∂Φ∂θω,

where:

Φ is the permanent magnet flux linkage.

θ is the rotor angle.

ω is the mechanical rotational speed.

The height h of the trapezoidal rate of change of flux profile
is derived from the permanent magnet peak flux.

Integrating ∂Φ∂θ over the range 0 to π/2,

Φmax=h2(θF+θW),

where:

Φmax is the permanent magnet
flux linkage.

h is the rate of change of flux profile height.

θF is the rotor angle range over
which the back emf that the permanent magnet flux induces in the stator is
constant.

θW is the rotor angle range over
which back emf increases or decreases linearly when the rotor moves at
constant speed.

Rearranging the preceding equation,

h=2Φmax/(θF+θW).

Electrical Defining Equations

Voltages across the stator windings are defined by

[vavbvc]=[Rs000Rs000Rs][iaibic]+[dψadtdψbdtdψcdt],

where:

va,
vb, and
vc are the external
voltages applied to the three motor electrical connections.

Rs is the equivalent resistance
of each stator winding.

ia,
ib, and
ic are the currents
flowing in the stator windings.

dψadt,dψbdt, and dψcdt

are the rates of change of magnetic flux in each stator winding.

The permanent magnet and the three windings contribute to the total flux linking
each winding. The total flux is defined by

[ψaψbψc]=[LaaLabLacLbaLbbLbcLcaLcbLcc][iaibic]+[ψamψbmψcm],

where:

ψa,
ψb, and
ψc are the total fluxes
linking each stator winding.

Laa,
Lbb, and
Lcc are the
self-inductances of the stator windings.

Lab,
Lac,
Lba, etc. are the mutual
inductances of the stator windings.

ψam,
ψbm, and
ψcm are the permanent
magnet fluxes linking the stator windings.

The inductances in the stator windings are functions of rotor angle, defined by

Laa=Ls+Lmcos(2θr),

Lbb=Ls+Lmcos(2(θr−2π/3)),

Lcc=Ls+Lmcos(2(θr+2π/3)),

Lab=Lba=−Ms−Lmcos(2(θr+π/6)),

Lbc=Lcb=−Ms−Lmcos(2(θr+π/6−2π/3)),

and

Lca=Lac=−Ms−Lmcos(2(θr+π/6+2π/3)),

where:

Ls is the stator
self-inductance per phase — The average self-inductance of each
of the stator windings.

Lm is the stator inductance
fluctuation — The amplitude of the fluctuation in self-inductance
and mutual inductance with changing rotor angle.

Ms is the stator mutual
inductance — The average mutual inductance between the stator
windings.

Parameterization for defining the permanent magnet flux distribution
as a function of rotor angle. Choose:

Perfect trapezoid - specify maximum flux
linkage to specify the maximum flux linkage
for the permanent magnet and the rotor angle where the back emf
is constant. The block assumes a perfect trapezoid for the back
emf. This is the default value.

Perfect trapezoid - specify maximum rotor-induced
back emf to specify the maximum rotor-induced
back emf and the corresponding rotor speed. The block assumes a
perfect trapezoid for the back emf.

Tabulated - specify flux partial derivative with
respect to rotor angle to specify values for
the partial derivative of flux linkage and the corresponding
rotor angles.

Tabulated - specify rotor-induced back emf as a
function of rotor angle to specify the
measured back emf constant and the corresponding rotor speed and
angles.

Vector of values for the rotor-induced back emf as a function of rotor
angle. The first and last values must be the same, and are normally both
zero. For more information, see the Corresponding rotor
angles parameter. First and last values are the same
because flux is cyclic with period 2π/N, where N is the number of permanent magnet pole
pairs.

Dependencies

This parameter is visible only when you set the Back EMF
profile parameter to Tabulated - specify
rotor-induced back emf as a function of rotor
angle.

Vector of values for the partial derivative of flux linkage (where
flux linkage is flux times number of winding turns) with respect to
rotor angle. The first and last values must be the same, and are
normally both zero. For more information, see the
Corresponding rotor angles parameter. First and
last values are the same because flux is cyclic with period 2π/N, where N is the number of permanent
magnet pole pairs.

Vector of rotor angles where the flux linkage partial derivative or
rotor-induced back emf is defined. Rotor angle is defined as the angle
between the a-phase magnetic axis and the
d-axis. That is, when the angle is zero, the
magnetic fields due to the rotor and the a-phase
winding align. This definition is used regardless of your block setting
for rotor angle definition. The first value is zero, and the last value
is 2π/N, where N is the number of permanent
magnet pole pairs.

Dependencies

This parameter is visible only when you set the Back EMF
profile parameter to Tabulated - specify
flux partial derivative with respect to rotor
angle or to Tabulated - specify
rotor-induced back emf as a function of rotor
angle.

Rotor speed used for back emf measurement — Rotor speed used for back emf measurement600rpm (default)

Specify the rotor speed corresponding to the maximum rotor-induced
back emf.

Dependencies

This parameter is visible only when you set the Back EMF
profile parameter to Perfect trapezoid -
specify maximum rotor-induced back emf or
Tabulated - specify rotor-induced back emf as a
function of rotor angle.

Number of pole pairs — Number of pole pairs6 (default)

Number of permanent magnet pole pairs on the
rotor.

Zero sequence — Zero sequence optionInclude (default) | Exclude

Option to include or exclude zero-sequence terms.

Include — Include zero-sequence terms. To prioritize model fidelity, use this default setting. Using this option:

Rotor angle definition — Reference point for the rotor angle measurementAngle between the a-phase magnetic axis and
the d-axis (default) | Angle between the a-phase magnetic axis and the
q-axis

Reference point for the rotor angle measurement. The default value is
Angle between the a-phase magnetic axis and the
d-axis. This definition is shown in the Motor Construction figure. When you
select this value, the rotor and a-phase fluxes are
aligned when the rotor angle is zero.

The other value you can choose for this parameter is Angle
between the a-phase magnetic axis and the q-axis. When you
select this value, the a-phase current generates
maximum torque when the rotor angle is zero.

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